The solid angle of a circular sector specified by circle radius, angle of the sector, and distance of the circle plane to the observer is calculated in terms of inverse trigonometric functions and elliptic integrals of the third kind. This generalizes results that have previously appeared in the literature for the full circle.

The even and odd Zernike Polynomials R_n^m(x) can be expanded into sums of even and odd Chebyshev Polynomials T_i(x). This manuscript provides closed forms for the rational expansion coefficients c_n,m,i for a set of small 0 <= n-m <= 6 and a holonomic five-term recurrence for these coefficients for all larger n.

The manuscript studies configurations of non-overlapping non-bonding dominoes on finite rectangular boards of unit squares characterized by row and column number. The non-bonding dominoes are defined here by the requirement that any domino on the board shares at most one point (one of its four corner points) with any other domino, but no edge. With the Transfer Matrix Method, rational generating functions are derived that solve the enumeration problem entirely, here evaluated for boards with up to six rows or columns.

This paper presents an overview of the SDSS-V Local Volume Mapper (LVM) telescope system. LVM is one of three surveys that form the fifth generation of the Sloan Digital Sky Survey, and it employs a coordinated network of four, 16cm telescopes feeding three fiber spectrographs at the Las Campanas Observatory. The goal is to spectrally map more than 4000 square degrees of the Galactic plane with 37" spatial resolution and R similar to 4000 spectral resolution over the wavelength range 360-980 nm. This corresponds to roughly 50 million individual spectra, which will reveal how distinct gaseous environments within our Galaxy interact with the stellar population, producing the large-scale interstellar medium that we observe. Accurately mapping and calibrating a substantial portion of the sky in this way requires a unique type of telescope. Each of the four units consists of a two-mirror siderostat in alt-alt configuration feeding an optical breadboard. This produces a fixed, stable focal plane for the fiber IFU and bundle. One telescope hosts the science IFU, while two others observe adjacent dark fields to calibrate geocoronal emission. The fourth telescope makes rapid observations of bright stars to compensate telluric absorption. The entrance slits of the spectrographs intersperse the fibers from all three types of telescope, producing truly simultaneous science and calibration exposures. After roughly four years of design, development, construction, testing, and commissioning, the LVM telescopes entered regular survey operations in late 2023. We summarize the entire LVM telescope project, from input scientific requirements to the actual performance achieved on-sky.

This paper presents the Local Volume Mapper (LVM) telescope system. LVM is one of three surveys that form the fifth generation of the Sloan Digital Sky Survey (SDSS-V), and it employs a coordinated network of four, 16 cm diameter telescopes feeding integral field units (IFUs) leading to three fiber spectrographs. One telescope hosts the science IFU, while two others observe adjacent fields to calibrate geocoronal emission. The fourth telescope makes rapid observations of bright stars to compensate for telluric absorption. The entrance slits of the spectrographs intersperse the fibers from all three types of telescope, producing truly simultaneous science and calibration exposures. We describe the final design of the telescope system and report on its construction, alignment and testing in the laboratory, as well as the integration, testing, commissioning, and actual on-sky performance at Las Campanas Observatory.

We describe the Sloan Digital Sky Survey Local Volume Mapper Instrument (LVM-I) construction, testing, and initial performance. The facility is designed to produce the first integral map of thousands of degrees of the Southern sky. The map will cover spectra from bluer than [O II] to 980 nm with a dispersion of over R =Delta lambda/lambda > 4, 000 at Ha wavelength. Each spaxel will have a pitch of similar to 35 '', and the survey will be conducted using four integral field units (IFUs) with an instantaneous field of view of 530 arcmin(2). The LVM facility is designed to achieve the required sub-Rayleigh spectroscopy over large sky areas with outstanding spectrophotometric accuracy and precision. LVM-I is designed to produce this unique dataset using four siderostats on commercial mounts. The four beams are fed into 16-cm-diameter f/11.4 apochromatic objectives, and the sky is derotated with K mirrors. These telescopes produce an image of the field onto both guider cameras and a lenslet array. The array reimages the field at f/3.7 onto 107-micron-diameter fibers. Blue throughput is maximized with a short 18.5-m fiber run from the IFUs to the spectrographs. The fibers are reconfigured inside a splicing box to distribute the fibers from the four telescopes to three spectrographs. The spectrographs are near-copies of the Dark Energy Survey three-band f/1.7 spectrographs, which deliver sharp images over the entire chromatic range. Nine STA charge-coupled devices (CCDs), cooled with liquid-nitrogen dewars, are used for the survey. The LVM-I is controlled with custom Python software and distributed over various computers using power-over-ethernet networking. The system is housed in a custom enclosure with a roll-off roof to grant access to the sky. The enclosure allows all four telescopes to point all over the sky and measure the transmissivity of the atmosphere and the sky background. Some of the first-light data products are highlighted here.

The double sum sum_(j=0)^m sum_(i=0)^j (-1)^(j-i) C(m,j) C(j,i) C(j+k+qi,j+k) with free nonnegative integer parameters k and q is rewritten as hypergeometric series. Efficient formulas to generate the C-finite ordinary generating functions are presented.

The definite integral with the kernel x/(x^2+b^2)/[\exp(2\pi x)-1] integrated from x=0 to infinity is the main term of a representation of the Digamma-Function psi(b), the derivative of the logarithm of the Gamma-Function. We present relations within the set of integrals over x^n/(x^2+b^2)^j/[\exp(\mu x)-1]^s for small integer exponents n, j and s with the aim to reduce them all to Polygamma-Functions.

The Foucault Pendulum is a Spherical Pendulum of fixed length with two angular degrees of freedom, attached to a suspension which rotates once a day around the Earth axis at a distance essentially set by Earth radius and the geodetic latitude of the pendulum. We write the Lagrange Function in the inertial frame of the fixed Earth axis and couple it strictly to the rotating frame at which the suspension appears at rest. The Euler-Lagrange equations are a coupled system of second-order differential equations for the two coordinates of the mass projected on the local horizontal plane. These are solved numerically in a C++ program which allows to study the trajectories beyond the various standard approximations of the literature.

This paper presents an update on the construction, testing, and commissioning of the SDSS-V Local Volume Mapper (LVM) telescope system. LVM is one of three surveys that form the fifth generation of the Sloan Digital Sky Survey, and it will employ a coordinated network of four, 16-cm telescopes feeding three fiber spectrographs at the Las Campanas Observatory. The goal is to spectrally map approximately 2500 square degrees of the Galactic plane with 37” spatial resolution and R~4000 spectral resolution over the wavelength range 360-980 nm. LVM will also target the Magellanic Clouds and other Local Group galaxies. Each of the four LVM telescopes consists of a two-mirror siderostat in alt-alt configuration feeding an optical breadboard. This produces a fixed, stable focal plane for the fiber-based Integral Field Unit (IFU). One telescope hosts the science IFU, while two others observe adjacent fields to calibrate geocoronal emission. The fourth telescope makes rapid observations of bright stars to compensate telluric absorption. The entrance slits of the spectrographs intersperse the fibers from all three types of telescope, producing truly simultaneous science and calibration exposures. We summarize the final design of the telescope system and report on its construction, alignment and testing in the laboratory. We also describe our deployment plan for commissioning at LCO, anticipated for late 2022.

The manuscript provides formulas for the volume of a body defined by the intersection of a solid cone and a solid sphere as a function of the sphere radius, of the distance between cone apex and sphere center, and of the cone aperture angle. If the sphere center lies on the (extended) cone axis the analysis may be based on cylinder coordinates fixed at the cone axis, and the volume is the sum of the well-known volumes of finite cones and sphere caps. At the general geometry the sphere center is not on the (extended) cone axis. Our approach calculates the volume by slicing space perpendicular to the cone axis and by integrating the lens areas defined by the sphere-cone intersection. These volume integrals are rephrased with the aid of the Byrd-Friedmann tables to Elliptic Integrals of the First, Second and Third Kind.

We derive the P-finite recurrences for classes of sequences with ordinary generating function containing roots of polynomials. The focus is on establishing the D-finite differential equations such that the familiar steps of reducing their power series expansions apply.

LINC-NIRVANA at the LBT has a dual MCAO system using solely natural guide stars. A multi pyramid WFS provides the slopes to close two independent loops for ground and high layer correction, as foreseen by the Layer Oriented scheme. The projection of the deformable mirrors actuators pattern on the WFS cameras rotates, since mechanical (ground) and optical (high) derotation provides the field rotation correction needed to keep the WFS on the reference stars. We reported in the previous conferences that we succeed in obtaining valid control matrices through numerical interpolation of a few calibrated interaction matrices registered for different clocking angle. We successfully tested a different approach based on a synthetic WFS and DM model. We proved that control matrices computed by the inversion of the interaction matrices generated from the model were effectively working on the real system closing high-order correction loop on the sky, providing better performance.

PANIC1 , the PAnoramic Near-Infrared Camera for the Calar Alto Observatory in Spain, was successfully commissioned in late 2014 with a mosaic of four 2K x 2K HAWAII-2RG arrays covering a field of view of approximately 30 arcmin at the 2.2 m telescope. Unfortunately, two of its science detectors suffered from extreme degradation2 along the years of operation making the instrument unsuitable for science observations. In the light of new technologies and constant innovations, it was decided to upgrade the instrument with a monolithic state of the art 4K x 4K HAWAII-4RG array. With as minimum mechanical and optical impact as possible for the instrument, the MPIA is the responsible institute for the challenging upgrade. Besides presenting the results of the initial operation of the HAWAII-4RG array in 64-Channel mode, the newly in-house designed detector mount will also be highlighted. In order to take as much advantage as possible of the new detector readout capabilities, and thanks to the modularity and flexibility of the in-house readout electronics, all 64 channels of the detector are read out in parallel. This allows for shorter integration times, which is very advantageous for a wide field imager with high background conditions.

This report models the refractive index above a telescope site by an atmosphere with exponential decay of the refractive index (susceptibility) as a function of altitude. The air is represented as a spherical hull around the surface. We compute (i) the differential zenith angle -- the difference between the actual zenith angle at arrival of rays at the telescope and a hypothetical zenith angle without the atmosphere -- and (ii) the optical path length distribution of rays at arrival in the entrance pupil as a function of the distance to the pupil center. The key technique in this work is to expand some integrals -- that depend on the refractive index profile along the curved path of each ray from some virtual plane in the direction of the star up to the entrance pupil -- in power series of small parameters.

We consider incomplete tilings of the equilateral triangle of edge length n that is subdivided into n^2 regular equilateral smaller unit triangles. Pairs of the unit triangles that share a side may be converted into lozenges, leaving some subset of the unit triangles untouched. We count numerically these coverings by lozenges and unit triangles for edge lengths n <= 6: the total and the detailed refinement as a function of the number of lozenges.

The manuscripts tabulates arc lists of the 1, 1, 3, 8, 25, 85, 397 ... unlabeled 2-regular digraphs on n=0, 1, 2, ..., 9 nodes, including disconnected graphs, graphs with multiarcs and/or graphs with loops. Each of these graphs represents one term of the Lagrangian of Lovelock's type -- a contraction of a product of n Riemann tensors -- once the 2 covariant and 2 contravariant indices of a tensor are associated with the in-edges and out-edges of a node.

The CARMENES radial velocity (RV) survey is observing 324 M dwarfs to search for any orbiting planets. In this paper, we present the survey sample by publishing one CARMENES spectrum for each M dwarf. These spectra cover the wavelength range 520--1710nm at a resolution of at least $R > 80,000$, and we measure its RV, H$\alpha$ emission, and projected rotation velocity. We present an atlas of high-resolution M-dwarf spectra and compare the spectra to atmospheric models. To quantify the RV precision that can be achieved in low-mass stars over the CARMENES wavelength range, we analyze our empirical information on the RV precision from more than 6500 observations. We compare our high-resolution M-dwarf spectra to atmospheric models where we determine the spectroscopic RV information content, $Q$, and signal-to-noise ratio. We find that for all M-type dwarfs, the highest RV precision can be reached in the wavelength range 700--900nm. Observations at longer wavelengths are equally precise only at the very latest spectral types (M8 and M9). We demonstrate that in this spectroscopic range, the large amount of absorption features compensates for the intrinsic faintness of an M7 star. To reach an RV precision of 1ms$^{-1}$ in very low mass M dwarfs at longer wavelengths likely requires the use of a 10m class telescope. For spectral types M6 and earlier, the combination of a red visual and a near-infrared spectrograph is ideal to search for low-mass planets and to distinguish between planets and stellar variability. At a 4m class telescope, an instrument like CARMENES has the potential to push the RV precision well below the typical jitter level of 3-4ms$^{-1}$.

We construct an orthogonal basis of functions defined over the unit circle as the product of the common sinusoidal functions of the azimuth angle by radial functions which are essentially sines of a polynomials of the radial distance to the origin. The main impetus of this approach is to generate basis functions where the minima and maxima along both coordinates, the azimuth and the distance r to the center, have the same amplitude, akin to the Chebyshev polynomial basis of the one-dimensional unit interval. The construction is based on numerical evaluation of the overlap integrals, which have the format of generalized Fresnel integrals.